[0001] The present invention relates to an image forming apparatus such as a digital copy
machine, which reads an image on an original sheet, processes the read image by an
image processing device, and forms the image on a sheet.
[0002] Recently, image forming apparatus such as digital copy machines and facsimile devices
have been rapidly popularized. In these digital office apparatus, the most important
operation is the image processing for creating a high-quality image output from an
image input by a scanner. Generally, the image input/output apparatus such as digital
copy machines and facsimile devices, entail the problems of various types of deteriorations
of space frequency as follows: the deterioration of the MTF (modulation transfer function)
characteristics of the optical system for reading, and the generation of folded distortion
caused by the limitation of the frequency band for digital sampling, which take place
during the data inputting; the deterioration in the developing system; the generation
of moire which is unique to digital process, and the like.
[0003] In order to avoid these problems, the image processing device requires an MTF correction.
Such an MTF correction is called "filtering process", which is further divided into
a low pass filtering (to be abbreviated as LPF) for suppressing moire and the like,
and a high pass filtering (to be abbreviated as HPF) for emphasizing an edge or the
like, of a character.
[0004] Conventionally, the LPF and HPF are carried out one at a time in series in the course
of the flow of the entire image process, that is, an input image is subjected first
to the LPF, then to the range correction process for correcting the contrast of the
image, and to the HPF.
[0005] In general, the filtering process of a digital copying machine, facsimile or the
like, is performed through a two-dimensional digital filtering process in the main
scanning and the sub-scanning directions. More specifically, the filtering process
is performed by multiplying each pixel of a local region consisting of a pixel to
be processed (focused pixel) and peripheral pixels, by a coefficient which corresponds
to a respective pixel.
[0006] Therefore, in the case where a process is performed by a mask size of (n × n), the
process in the sub-scanning direction requires line buffers of n lines. However, the
filtering process is carried out by the conventional method, the process is performed
in a serial manner, and therefore in the case of the LPF and HPF process of a mask
size of (3 × 3), line buffers of a total of 4 lines, two lines for each, are required.
[0007] As described above, with the conventional technique, in the case where a filtering
process of an image processing apparatus is carried out by a mask size of (n × n),
the process in the sub-scanning direction requires line buffers of n lines. With this
structure, if the filtering process is carried out by the conventional method, the
process is performed in a serial manner, and therefore in the case of the LPF and
HPF process of a mask size of, for example, (3 × 3), line buffers of a total of 4
lines, two lines for each, are required.
[0008] The object of the present invention is to provide an image forming apparatus capable
of reducing the number of line buffers used in the filtering process carried out by
a mask size of (n × n).
[0009] According to the present invention, there is provided an image forming apparatus
including: means for reading image data from an original sheet; first processing means
for processing the image data read by the reading means through a low pass filter,
and outputting a first process result; second processing means, provided in parallel
with the first processing means, for processing the image data read by the reading
means through a high pass filter having a filter frequency range higher than that
of the low pass filter, and outputting a second process result; adding means for adding
up the first process result and the second process result, and outputting a result
of addition; and image forming means for forming an image on the basis of the result
of addition.
[0010] With the above-described structure, the present invention exhibits the following
advantage. That is, the present invention does not employ the structure of the conventional
technique, in which a plurality of filtering processes are arranged in series, and
each filtering process is carried out one at a time in sequence, but according to
the present invention, a plurality of filtering processes are carried out in a parallel
manner for image data. With the structure of the present invention described above,
it is no longer necessary to provide a line buffer for each and every filter as in
the conventional technique, and therefore the size and the production cost of the
apparatus can be reduced.
[0011] More specifically, with the conventional technique, for example, 2 lines of line
buffers are required just before each filter, and therefore when there are two filters,
a total of 4 lines of line buffers are required. By contrast, with the structure of
the present invention, image data are processed in parallel, and therefore it suffices
only if line buffers of the number for only one filter in the conventional case should
be provided. Consequently, in the case where there are two filters, the same performance
can be achieved by only 2 lines of line buffers. For this reason, it becomes possible,
with the structure of the present invention, to reduce the size and production cost
of the apparatus.
[0012] This invention can be more fully understood from the following detailed description
when taken in conjunction with the accompanying drawings, in which:
FIG. 1 is a block diagram illustrating the structure of the image processing unit
in a digital copying machine according to the present invention;
FIG. 2 is a cross sectional view of the brief structure of the digital copying machine,
which includes the image processing apparatus of the present invention;
FIG. 3 is a schematic diagram showing the structure of the control system of the digital
copying machine shown in FIG. 2;
FIG. 4 is a diagram showing an example of the circuit of the image processing unit
shown in FIG. 1;
FIG. 5 is a table containing the values of pixels within a local region (3 × 3), of
the image data;
FIG. 6 is a table containing the coefficients for low pass filters;
FIG. 7 is a table containing the coefficients for high pass filters;
FIG. 8 is a table showing results of the evaluation of images in the case where the
value of parameter K is varied; and
FIGS. 9A, 9B and 9C are tables each containing
[0013] An embodiment of the present invention will now be described with reference to accompanying
drawings.
[0014] FIG. 2 is a cross sectional view of an example of the internal structure of a digital
copying machine, which is an embodiment of the image forming apparatus according to
the present invention.
[0015] As can be seen in the figure, the digital copying machine includes a scanner unit
1 for optically reading image data on an original O, and a printer unit 2 for outputting
an image read via the scanner unit 1 on a recording medium, which is a copy sheet
P.
[0016] In the scanner unit 1, an original to be copied, is placed on an original placement
table 3, and the original O placed on the original placement table 3, is irradiated
by a fluorescent lamp 4 extended in the sub-scanning direction, which serves as a
light source. The light beam reflected from the original O irradiated by the fluorescent
lamp 4 is photoelectrically converted by a CCD sensor 5 which serves as a photoelectric
conversion element, and thus the image data of the reflection light beam is converted
into an image signal.
[0017] All along the fluorescent lamp 4, a reflector 6 is provided so as to converge the
illumination light from the fluorescent lamp 4 on the original O at a high efficiency.
Further, between the fluorescent lamp 4 and the CCD sensor 5, a plurality of mirrors
for bending optical paths through which light beams reflected from the original O
pass, which proceed towards the CCD sensor 5 from the original O, a lens 10 for converging
the reflection light beams on the light converging surface of the CCD sensor 5, and
the like are provided.
[0018] The optical system for guiding the reflection light beams from the exposure lamp
4 and the original O to the CCD sensor 5 is provided on the carriages 11 and 12, and
are conveyed in the main scanning direction by means of a pulse motor which is not
shown. The sub-scanning directional area of the original O is irradiated by the fluorescent
lamp 4, and the sub-scanning directional regions of the original O is sequentially
irradiated as the carriages 11 and 12 are conveyed in the main scanning direction.
Thus, the entire region of the original O is irradiated by the fluorescent lamp 4.
[0019] On the top of the original placement table 3, a original sheet cover 13 is provided
for bringing the original O into tight contact with the placement table 3. The original
sheet press cover can be replaced by, for example, an SDF, a semi-automatic original
sheet feeding device, or an ADF, an automatic original sheet feeding device, in accordance
with the size or performance of the digital copying machine.
[0020] In the printer unit 2 serving as the image forming device, a cylindrical photosensitive
drum 14 which can be rotated in a desired direction by means of, for example, a motor
(not shown) and is charged at a pre-determined potential, is provided. A laser beam
is applied on the photosensitive drum 14, the potential of the beam-applied region
on the drum, is varied, and a static latent image is formed on the photosensitive
drum 14.
[0021] As surroundings of the photosensitive drum 14, there are arranged a charging device
15 for applying a predetermined potential to the photosensitive drum 14, a laser unit
16 for outputting a laser beam modulated in accordance with the image data, on the
photosensitive drum 14, a developing device for developing a static latent image formed
on the photosensitive drum 14 by the laser beam output from the laser unit 16, by
supplying a visualizer, that is, toner, to the image, a transfer device 18 for transferring
a toner image visualized on the photosensitive drum 14 by development by the developing
device 17, on a recording medium, that is, a copy sheet P which is fed from a recording
medium feeding unit, which will be explained later, and a removing device 19 for removing
the copy sheet P from the photosensitive drum 14.
[0022] The laser unit 16 consists of a semiconductor laser oscillator for generating a laser
beam, a polygon mirror 25 for converting the laser beam supplied from the semiconductor
laser oscillator 24 via a collimate lens (not shown) into beams of scanning lines,
each beam for one scanning line, an fθ lens 26 for converting the laser beams of scanning
lines from the polygon mirror 25, into parallel light beams, a mirror 27 for reflecting
the parallel light beams from the lens 26 so as to guide them to the photosensitive
drum 14, and a mirror motor 28 for rotating the polygon mirror 25.
[0023] In addition, on the down-stream side of the removing device 19 with respect to the
rotational direction of the photosensitive drum 14, a cleaner unit 20 is provided
for the purpose of removing toner remaining on the surface of the photosensitive drum
14, and erasing the variance in potential created on the photosensitive drum 14 by
the laser beam, for the next image forming operation.
[0024] Between the developing device 17 and the transfer device 18, a recording medium feeding
unit 21 for feeding a copy sheet P on which a toner image formed on the photosensitive
drum 14 is to be transferred, towards the transfer device 18, is provided.
[0025] Further, at a position along the direction in which the copy sheet P having a toner
image transferred by the transfer device 18, is separated from the photosensitive
drum 14, a fixing device 22 for fixing the toner image on the copy sheet, is provided.
Between the fixing device 22 and the transfer device 18, a conveying device 23 for
conveying copy sheets P to the fixing device 22, is provided.
[0026] FIG. 3 is a block diagram showing an example of the control system of the copying
machine shown in FIG. 2.
[0027] As shown in FIG. 3, a calculation unit 44 for performing various types of calculations,
and four sub-control units 32 to 34 each for executing the control of a respective
unit, are connected to a main control unit 31 which controls the overall operation
of the digital copying machine.
[0028] The main control unit 31 is connected to an operation panel 35 for instructing various
types of image processing, a region management unit 36 for managing an image processing
region, and an image processing unit 37 serving as an image processing device for
improving the quality of image data inputted, editing the image data, and processing
the image data, and the control unit 31 serves to control these members.
[0029] The sub-control unit 32 is connected to a light source control unit 40 for controlling
the source light intensity of the fluorescent lamp 4, a mechanism driving unit 42
for controlling a mechanical input mechanism 41 such as the sheet feeding unit shown
in FIG. 2, and an A/D conversion unit 43 for converting analog image data converted
by the CCD sensor 5 for converting detected reflection light into image data, into
a digital data signal, and the sub-control unit 32 serves to control these members.
[0030] The sub-control unit 33 is connected to an image expanding unit 45 for expanding
edited or processed image data, for forming an image, and also storing the data, an
image output unit 46 for outputting image data from the image expanding unit 45 (image
data from the scanner unit 1) in the form of a pulse signal as a laser modulation
signal (image data for the printer unit 2), and a mechanism drive unit 49 for driving
an output unit mechanism 48 including drive systems such as motor and solenoid in
the laser unit 16, and the sub-control unit 33 serves to control these members.
[0031] The sub-control unit 34 is connected to a data transmission/reception unit 50, and
the unit 34 controls the transmission/reception of data with regard to an external
devices.
[0032] In the digital copying machine described above, an original O is illuminated by the
fluorescent lamp 4, and the reflection light reflected from the original O forms an
image on the CCD sensor 5, where the image is converted into an analog electric signal.
The analog image signal is converted into a digital signal by the A/D converting unit
43, and the digital signal converted is output to the image processing unit 37.
[0033] The image processing unit 37 performs, for example, the process of filtering image
data from the scanner unit 1, the enlargement or contraction of image data in the
main scanning direction, and the gradation process.
[0034] FIG. 1 shows the internal structure of the image processing unit 37 serving as the
image processing apparatus.
[0035] The image processing unit 37 includes a low pass filter (to be called LPF hereinafter)
51 for suppressing moire or the like, of image data read by the scanner unit 1, a
range correction unit 52 for correcting the irregularity of the CCD sensor 5, a high
pass filter (to be called HPF hereinafter) 53, connected in parallel with the LPF
51, for emphasizing an edge or the like, of a character of the image read by the scanner
unit 1, a multiplier for multiplying image data from the range correction unit 52,
with a parameter K, which will be described later, an adder 55 for adding image data
from the multiplier 54 to image data from the range correction unit 52, an enlargement/contraction
unit 56 for subjecting the image data from the adder 55 to an enlargement or contraction
process, and γ-correction unit 57 for performing various types of γ-corrections on
the image data, and an error diffusion processing unit 58 for subjecting image data
from the γ-correction unit 57 to an error diffusion process.
[0036] FIG. 4 shows an example of the circuit for performing the process of the LPF 51 and
HPF 52 for the mask size (3 × 3) of the structure of the image processing unit 37
shown in FIG. 1. It should be noted that all of the data handled in this case are
of 8-bit.
[0037] More specifically, in the image processing unit 37 for a mask size of (3 × 3), two
line buffers, namely, a line buffer (LD1) 61 for taking image data in from the scanner
unit 1, and a line buffer (LD2) 62 are provided. Further, the range correction unit
52 is connected to a range correction reference correction value calculating unit
60 for calculating out a range correction reference value from the image data from
the scanner 1.
[0038] Next, the operation of the image processing unit 37 of the present invention will
now be described with reference to the circuit example of mask size (3 × 3) shown
in FIG. 4.
[0039] First, image data input from the scanner 2 is temporarily taken in the line buffers
61 and 62.
[0040] Supposing, of the image data input in synchronous with predetermined clocks from
the line buffers 61 and 62, the values of the pixels of a local region of (3 × 3)
are f(i-1, j-1), f(i, j-1), f(i+1, j-1), f(i-1, j), f(i, j), f(i+1, j), f(i-1, j+1),
f(i, j+1) and f(i+1, j+1), the result, that is, l(i, j), of the process of the focused
pixel, f (i, j), by the LPF 51 can be expressed by the following equation:

where La, Lb, Lc and Ld are coefficients for the LPF 51 of (3 × 3) shown in FIG.
6, and they must satisfy the following condition in order not to cause a change in
the local average concentration between before and after the process:

[0041] Next, the result of the process by the LPF 51 is further subjected to the range correction
process by the range correction unit 52 for the purpose of correcting the contrast
of the image, and let us suppose the value of this process is r (i,j).
[0042] Further, in parallel with the process by the LPF 51 and the range correction process
of the range correction processing unit 52, the process by the HPF 53 is carried out
in a manner similar to that of the LPF 51 as follows. That is, of the image data taken
in by the line buffers 61 and 62, the values of the pixels of a local region of (3
× 3), that is, f(i-1, j-1), f(i, j-1), f(i+1, j-1), f(i-1, j), f(i, j), f(i+1, j),
f(i-1, j+1), f(i, j+1) and f(i+1, j+1), are multiplied with coefficients Ha, Hb, Hc
and Hd.
[0043] To be specific, the result, that is, h(i, j), of the process of the focused pixel,
f (i, j), by the LPF 53 can be expressed by the following equation:

[0044] The coefficients Ha, Hb, Hc and Hd of the HPF 53 are arranged as can be seen in FIG.
7. Further, the intensity of emphasizing the edge can be changed by multiplying the
result of the process by the HPF 53 with the parameter K. In other words, as the value
of the coefficient is higher, the degree of the emphasizing the edge becomes higher,
and therefore the optimal value can be set in accordance with the MTF characteristics
and the like, of the scanner unit 2.
[0045] The results of the calculations performed in parallel are added up on the basis of
the following equation, and the result, f(i, j) is subjected to the processes from
the enlargement/contraction process onwards, shown in FIG. 1. Then, the processed
result is output by the printer.

[0046] With the method discussed above, the filtering process of pixels having a mask size
of (3 × 3) can be realized by only 2 lines of buffers.
[0047] It should be noted that, in the case where the process is carried out through the
conventional method, the process of the HPF is carried out on the result of the range
correction, and therefore 2 lines of buffers for temporarily storing the result of
the range result, and another 2 lines of buffers for reading an image input via the
scanner 2 at the beginning, a total of 4 lines are required.
[0048] By contrast, in the present invention, the processes by the LPF 51 and the HPF 53
are performed in parallel, and therefore it becomes possible to reduce the number
of line buffers by 2.
[0049] Further, in the conventional technique, the result of the processing of an input
image by the LPF 51, or by the range correction unit 52, is then subjected to the
process by the HPF 53, whereas in the present invention, an input image is subjected
directly to the process by the HPF 53, and therefore the degree of the emphasis of
the edge is made stronger as compared to the conventional technique. As a result,
there may rise a problem of easily generating a moire; however such a problem can
be solved by setting the parameter K used to change the intensity of the process by
the HPF 53, to the optimal value.
[0050] FIG. 8 shows the results of the evaluation of an image in the case where the value
of the parameter K is varied. As shown in this table, with the conventional technique,
a good fine character reproducibility and a good moire suppression property were obtained
when K = 1.3, whereas with the present invention, the edge emphasis is excessively
high when K = 1.3. A simulation test proved that when the K value is decreased to
about 0.6, an image of a quality as high as that of the conventional technique could
be obtained.
[0051] It should be noted that in the present invention, the variation of the image is slightly
large with respect to the variation of the value of the parameter K, and therefore
it is necessary to set the accuracy of the parameter K more finely than in the conventional
case.
[0052] In the embodiment described above, the coefficients listed in FIG. 9A were used for
the LPF 51, and those of FIG. 9B were used for the HPF 53.
[0053] Further, although the value, K = 0.6, is preferable as the parameter K, in terms
of a fine character reproducibility and the prevention of generating a moire, the
preferable value may vary slightly depending upon coefficients selected for the LPF
51 and the HPF 53.
[0054] For example, in the case where the coefficients for the HPF 53 are varied as can
be seen in FIG. 9C, the edge is emphasized more than that of the above-described embodiment,
and therefore the optimal value for the parameter K becomes a lower one. Thus, as
the value for the parameter K, an optimal value which corresponds to the coefficients
selected for the LPF 51 and the HPF 53 must be selected.
[0055] The above-described embodiment was discussed in connection with the case of a mask
size of (3 × 3); however the present invention is not limited to this mask size, but
the size can be freely set. Generally, in the case of a mask size of (n × n), the
number of line buffers necessary is n - 1, which is only a half as many of those required
in the conventional case.
[0056] Further, the arrangement of the coefficients is not limited to that discussed in
the above embodiment, and it is possible that all the coefficient values and it is
possible that all the coefficient values within a mask size are changed.
[0057] As described above, according to the embodiment of the present invention, the LPF
process and the HPF process for an image input from the scanner can be carried out
in parallel, and therefore the number of line memories can be reduced.
[0058] Further, when the value of the parameter K is optimized, an image of a quality as
high as that of the conventional case, can be obtained.
[0059] To summarize, the present invention provides an image forming apparatus capable of
reducing the number of line buffers used for the process with a mask size of (n ×
n) in the filtering process of the image processing device.
1. An image forming apparatus characterized by comprising:
means (1) for reading image data from an original sheet;
first processing means (51) for processing the image data read by the reading means
through a low pass filter, and outputting a first process result;
second processing means (53), provided in parallel with the first processing means,
for processing the image data read by the reading means through a high pass filter
having a filter frequency range higher than that of the low pass filter, and outputting
a second process result;
means (55) for adding up the first process result and the second process result, and
outputting a result of addition; and
means (2) for forming an image on the basis of the result of addition.
2. An image forming apparatus according to claim 1, characterized in that the first processing
means includes a range correction unit (52) for correcting an irregularity of the
image data on the basis of the first process result obtained by the first processing
means.
3. An image forming apparatus according to claim 1, characterized in that the second
processing means includes edge emphasizing intensity changing means for changing an
intensity of an edge emphasis by multiplying the second process result with a parameter
for adjusting the edge emphasis density.
4. An image forming apparatus according to claim 1, characterized by further comprising:
means (56) for contracting or enlarging the addition result obtained by the adding
means, and supplying the contracted or enlarged result to the image forming means.
5. An image forming apparatus according to claim 1, characterized by further comprising:
means (57) for performing a γ-correction on the addition result obtained by the adding
means, and supplying the corrected result to the image forming means.
6. An image forming apparatus according to claim 1, characterized by further comprising:
means (58) for correcting the addition result obtained by the adding means by means
of an error diffusion method, and supplying the corrected result to the image forming
means.
7. An image forming apparatus according to claim 1, characterized in that the first processing
means includes the low pass filter (51, FIG. 9A) having a size of 3 × 3, coefficients
of coordinates of which are: (0, 0) = (0.0625), (0, 1) = (0.1250), (0, 2) = (0.0625),
(1, 0) = (0.1250), (1, 1) = (0.2500), (1, 2) = (0.1250), (2, 0) = (0.0625), (2, 1)
= (0.1250) and (2, 2) = (0.0625); and
the second processing means includes the high pass filter (53, FIG. 9B) having
a size of 3 × 3, coefficients of coordinates of which are: (0, 0) = (1), (0, 1) =
(0), (0, 2) = (1), (1, 0) = (0), (1, 1) = (-4), (1, 2) = (0), (2, 0) = (1), (2, 1)
= (0) and (2, 2) = (1).
8. An image forming apparatus according to claim 1, characterized in that the first process.ing
means includes the low pass filter (51, FIG. 9A) having a size of 3 × 3, coefficients
of coordinates of which are: (0, 0) = (0.0625), (0, 1) = (0.1250), (0, 2) = (0.0625),
(1, 0) = (0.1250), (1, 1) = (0.2500), (1, 2) = (0.1250), (2, 0) = (0.0625), (2, 1)
= (0.1250) and (2, 2) = (0.0625); and
the second processing means includes the high pass filter (53, FIG. 9C) having
a size of 3 × 3, coefficients of coordinates of which are: (0, 0) = (1), (0, 1) =
(1), (0, 2) = (1), (1, 0) = (1), (1, 1) = (-8), (1, 2) = (1), (2, 0) = (1), (2, 1)
= (1) and (2, 2) = (1).
9. An image forming method characterized by comprising:
a reading step (1) for reading image data from an original sheet;
a first processing step (51) for processing the image data read by the reading means
through a low pass filter, and outputting a first process result;
a second processing step (53), provided in parallel with the first processing step,
for processing the image data read by the reading means through a high pass filter
having a filter frequency range higher than that of the low pass filter, and outputting
a second process result; and
an adding step (55) for adding up the first process result and the second process
result, and outputting a result of addition; and
an image forming step (2) for forming an image on the basis of the result of addition.
10. An image forming method according to claim 9, characterized in that the first processing
step includes a range correction step (52) for correcting an irregularity of the image
data on the basis of the first process result obtained by the first processing means.
11. An image forming method according to claim 9, characterized in that the second processing
step includes edge emphasizing intensity changing step (54) for changing an intensity
of an edge emphasis by multiplying the second process result with a parameter for
adjusting the edge emphasis intensity.
12. An image forming method according to claim 9, characterized by further comprising
a step (56) for contracting or enlarging the addition result obtained by the adding
means, and supplying the contracted or enlarged result to the image forming step.
13. An image forming method according to claim 9, characterized by further comprising
a step (57) for performing a γ-correction on the addition result obtained by the adding
means, and supplying the corrected result to the image forming step.
14. An image forming apparatus according to claim 9, characterized by further comprising
a step (58) for correcting the addition result obtained by the adding step by means
of an error diffusion method, and supplying the corrected result to the image forming
step.
15. An image forming method according to claim 9, characterized in that the first processing
step includes a step of using the low pass filter (51, FIG. 9A) having a size of 3
× 3, coefficients of coordinates of which are: (0, 0) = (0.0625), (0, 1) = (0.1250),
(0, 2) = (0.0625), (1, 0) = (0.1250), (1, 1) = (0.2500), (1, 2) = (0.1250), (2, 0)
= (0.0625), (2, 1) = (0.1250) and (2, 2) = (0.0625); and
the second processing step includes a step of using the high pass filter (53, FIG.
9B) having a size of 3 × 3, coefficients of coordinates of which are: (0, 0) = (1),
(0, 1) = (0), (0, 2) = (1), (1, 0) = (0), (1, 1) = (-4), (1, 2) = (0), (2, 0) = (1),
(2, 1) = (0) and (2, 2) = (1).
16. An image forming method according to claim 9, characterized in that the first processing
step includes a step of using the low pass filter (51, FIG. 9A) having a size of 3
× 3, coefficients of coordinates of which are: (0, 0) = (0.0625), (0, 1) = (0.1250),
(0, 2) = (0.0625), (1, 0) = (0.1250), (1, 1) = (0.2500), (1, 2) = (0.1250), (2, 0)
= (0.0625), (2, 1) = (0.1250) and (2, 2) = (0.0625); and
the second processing step includes a step of using the high pass filter (53, FIG.
9B) having a size of 3 × 3, coefficients of coordinates of which are: (0, 0) = (1),
(0, 1) = (1), (0, 2) = (1), (1, 0) = (0), (1, 1) = (-8), (1, 2) = (1), (2, 0) = (1),
(2, 1) = (1) and (2, 2) = (1).